Problem: Simplify. Multiply and remove all perfect squares from inside the square roots. Assume $z$ is positive. $\sqrt{z}\cdot\sqrt{30z^2}\cdot\sqrt{35z^3}=$
Let's start by merging the square roots: $\begin{aligned} \sqrt{z}\cdot\sqrt{30z^2}\cdot\sqrt{35z^3}&=\sqrt{z\cdot 30z^2\cdot 35z^3} \\\\ &=\sqrt{1050z^6} \end{aligned}$ Now we remove all perfect squares from inside the square root: $\begin{aligned} \sqrt{1050z^6}&=\sqrt{5^2\cdot 2\cdot 3 \cdot 7\cdot \left(z^3\right)^2} \\\\ &=\sqrt{5^2}\cdot\sqrt{42}\cdot\sqrt{ \left(z^3\right)^2} \\\\ &=5\cdot \sqrt{42}\cdot z^3 \\\\ &=5z^3\sqrt{42} \end{aligned}$ In conclusion, $\sqrt{z}\cdot\sqrt{30z^2}\cdot\sqrt{35z^3}=5z^3\sqrt{42}$